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Theorem ordtri4 4508
Description: A trichotomy law for ordinals. (Contributed by NM, 1-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordtri4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  ( A  C_  B  /\  -.  A  e.  B ) ) )

Proof of Theorem ordtri4
StepHypRef Expression
1 eqss 3270 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
2 ordtri1 4504 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
32ancoms 439 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
43anbi2d 684 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  C_  B  /\  B  C_  A )  <->  ( A  C_  B  /\  -.  A  e.  B ) ) )
51, 4syl5bb 248 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  ( A  C_  B  /\  -.  A  e.  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    C_ wss 3228   Ord word 4470
This theorem is referenced by:  carduni  7701  alephfp  7822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-tr 4193  df-eprel 4384  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474
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